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Several converses of the Banach contraction principle exist. The following is due to [[Bessaga]], from [[1959]]:
Let <math>f:X\rightarrow X</math> be a map of an abstract [[set (mathematics)|set]] such that each [[iterated function|iterate]] ''f''<sup>n</sup> has a unique fixed point. Let ''q'' be a real number, 0 < q < 1. Then there exists a complete metric on ''X'' such that ''f'' is contractive, and ''q'' is the contraction constant.
==Generalizations==
See the article on [[Fixed point theorems in infinite-dimensional spaces]] for generalizations.
==References==
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